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Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17]. [ABSTRACT FROM AUTHOR]

Given an analytic standardly normalized function g in D := {z ∊ C : ∣z∣ < 1}, by C(g) will be denoted the class of analytic standardly normalized function f such that Re {eiδzf'(z)/g(z)}>0, z ∊ D, for some δ ∊ (-π/2, π/2). For the class C(g) the Fekete-Szegö problem is examined. [ABSTRACT FROM AUTHOR]

The object of the present investigation is to solve Fekete-Szegö problem for a new class Vλp (a, c, A,B) of p-valent analytic functions involving the Saitoh operator in the unit disk. We also obtain subordination results and some interesting corollaries for functions in Ap involving this operator. Relevant connections of the results obtained here with those given by earlier workers on the subject are also mentioned. [ABSTRACT FROM AUTHOR]

We consider the space of analytic functions on the unit disk D, defined by the requirement fD f′(z) ø(z) dA(z) < ∞, where ø(r) = logα(1/(1 - r)) and show that it is a predual of the "logα-Bloch" space and the dual of the corresponding little Bloch space. We prove that a function with an ↓ 0 is in iffP∞ logα(n+2)/(n+1) < ∞ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in Some properties of the Cesàro and the Libera operator are considered as well. [ABSTRACT FROM AUTHOR]

The entire equitable dominating graph EEqD(G) of a graph G with vertex set V ∪ S, where S is the collection of all minimal equitable dominating sets of G and two vertices u, v Ε V ∪ S are adjacent if u, v are not disjoint minimal equitable dominating sets in S or u, v Ε D, where D is the minimal equitable dominating set in S or u G V and v is a minimal equitable dominating set in S containing u. In this paper, we initiate a study of this new graph valued function and also established necessary and sufficient conditions for EEqD(G) to be connected and complete. Other properties of EEqD(G) are also obtained. [ABSTRACT FROM AUTHOR]

Quasi-subordination is an underlying concept in the area of complex function theory. It is an interesting topic that unifies the concept of both subordination and majorization. There has been no work in this area for the past three decades except possibly a recent article (Haji Mohd and Darus, Fekete–Szegö problems for quasi-subordination classes, Abstr. Appl. Anal. 2012 (2012) 192956, 14 p.) [8] . Exploiting this article, we provide an estimate with k -th root transform for certain classes of analytic univalent functions using quasi-subordination. The authors sincerely hope that this article will revive this concept and encourage other researchers to work in this quasi-subordination in the near-future in the area of complex function theory. [ABSTRACT FROM AUTHOR]

This paper studies the recovery of the height field function of an object from its surface normal vectors in terms of uniform bi-quadratic B-spline surfaces. The study is motivated by two modern technological needs namely, the recovery of the height field function of an object from its surface normal vectors obtained by photometric stereo techniques, and the introduction of new design methodology of streamlined aesthetic surfaces through the direct editing of normal vectors. Our mathematical formulation for explicit surface reconstruction is based on differential geometry and geometric modeling techniques. Complex examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Let k - ST[A, B], k ≥ 0, -1 ≤ B < A ≤ 1 be the class of normalized analytic functions defined in the open unit disk satisfying R((B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)) > k/(B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)-1/ and let k - UCV [A,B], k ≥ 0, -1 ≤ B < A ≤ 1 be the corresponding class satisfying R((B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)) > k/(B-1)zf'(z)/f(z)-(A-1)/(B+1)zf'(z)/f(z)-(A+1)-1/. For an appropriate δ > 0, the δ neighborhood of a function f ∊ k -UCV[A, B] is shown to consist of functions in the class k - ST[A, B]. [ABSTRACT FROM AUTHOR]

The universality theoremasserts that the lower density of any set of shifts of the Riemann zeta-function which approximate a given analytic function with accuracy ɛ > 0 is strictly positive. It is proved that this set has strictly positive density for all but at most countably many ɛ > 0. [ABSTRACT FROM AUTHOR]

Applications of geometric function theory to some inequalities for algebraic polynomials and rational functions normalized on circular arcs are considered. In particular, coefficients estimates and also covering and distortion theorems are proved. The latter theorems supplement recent results of the author and V. N. Dubinin. [ABSTRACT FROM AUTHOR]

The dependence of values of functionals occurring in some extremal decomposition problems on the location of the poles of the associated quadratic differentials is established. The proof is based on an analysis of geometric properties of trajectories of two quadratic differentials. [ABSTRACT FROM AUTHOR]

Abstract: We prove here that the polynomial -enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition and has a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here is a rescaled Hall–Littlewood polynomial and ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of Haglund et al. This result includes as special cases all previous results connected with the shuffle conjecture such as the -Catalan, Schröder and two shuffle results of Haglund as well as their compositional refinements recently obtained by the authors. It also confirms the possibility that the approach adopted by the authors has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki. [Copyright &y& Elsevier]

Let π (x) be the number of primes not exceeding x. We prove that for , and that for sufficiently large We finally prove that for and k = 2, 3,..., 147297098200000, the closed interval [( k - 1) x, kx] contains at least one prime number, i.e. the Bertrand's postulate holds for x and k as above. [ABSTRACT FROM AUTHOR]

Hardy-type inequalities with power and logarithmic singularities are considered. New one-dimensional inequalities and their multidimensional analogs are proved. [ABSTRACT FROM AUTHOR]

Abstract: Following constructions of McMullen [11] and Vasilyev [15] we list strict quadratic differentials on (marked) tori defined by twists of one-forms on hyperelliptic surfaces. We count the total number of twists in each genus and show that the quadratic differentials on (marked) tori are strict. Taking universal covers we obtain a variety of (non-arithmetic) lattice Panov planes [14] from genus two lattice surfaces. [Copyright &y& Elsevier]

We investigate convolution properties and coefficients estimates for two classes of analytic functions involving the q-derivative operator defined in the open unit disc. Some of our results improve previously known results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to determine the conditions of starlikeness for certain class of multivalent analytic functions. Relevant connections of the results presented here with those obtained in earlier works are pointed out. [ABSTRACT FROM AUTHOR]

Abstract: For an analytic function on the open unit disk and continuous on , the Cantor boundary behavior (CBB) is used to describe the curve that forms infinitely many fractal-look loops everywhere. The class of analytic functions with the CBB was formulated and investigated in Dong et al. [6]. In this note, our main objective is to give further discuss of the criteria of CBB in Dong et al. [6]. We show that the two major criteria, the accumulation of the zeros of near the boundary and the fast mean growth rate of near the boundary, do not imply each other. Also we make an improvement of another criterion, which allows us to have more examples of CBB. [Copyright &y& Elsevier]

We first define a generalized form of a Hurwitz-Lerch zeta type function and then use it in constructing certain classes of analytic functions in the unit disk. In our investigation, we obtain various results for the classes introduced, thereby, exhibiting their useful properties and characteristics by adopting the techniques of differential subordination. Several consequences of the main results are considered and relevant connections with some of the known results are also pointed out. [ABSTRACT FROM AUTHOR]

We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Dirichlet L-functions. Such bounds require new techniques, which is visible from the relationship to nonvanishing results for . [ABSTRACT FROM AUTHOR]

We derive analytical solutions of the Schrödinger wave equation for some q-deformed potentials. Most of these solutions are obtained in terms of Heun functions. However, for some particular cases the solutions are obtained in terms of geometric and hypergeometric functions. Different observations are realized for each potential. However, for all the cases which we study the oscillations are the usual ones. It is also noted that all of these functions are sensitive to variation in the parameters involved. Our discussion for some particular potential has shown us that for a large value of the parameter q the system tends to exhibit the energy for an infinite potential well between two points zero and a in addition to a free particle. [ABSTRACT FROM AUTHOR]

We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(Ḡ) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p - 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m? [ABSTRACT FROM AUTHOR]

Abstract: The aim of this investigation is to reveal some relationships between certain multivalently analytic functions and multivalently meromorphic functions, and then to focus on certain consequences of these which will be important or interesting for analytic or geometric function theory. [Copyright &y& Elsevier]

Let S be the class of functions f(z)=z+ a2 z2 + ...,f(0)=0 , f'(0)=1 which are regular and univalent in the unit disk ∣z∣<1. For -1⩽-a<0< a ⩽1,a>0 we consider the equation: Re [(X2 - a2 ) f(x) ] = 0 , f ∊S , x∊[-1,1] . Denote φ(x)= Re [(X2 - a2 ) f (x) ]. Because φ(0)=0 and φ( ± a)=0 it follows that there is y∊ (-a,0) such that φ'( y)=0 and z∊(0, a) such that φ(z) = 0. The aim of this paper is to find min{y\ φ'( y)=0}and max{z\ φ'( z)=0}[3]. If y and z is min{y\ φ'(y)=0}, respectively max{z\ φ'(z)=0} then for x< y and x> z the equation φ'( x)=0 does not have real roots. Since S is a compact class, there exists . This problem was first proposed by Petru T. Mocanu .We will determine by using the variational method of Schiffer-Goluzin. [ABSTRACT FROM AUTHOR]

A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in D makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions. [ABSTRACT FROM AUTHOR]

Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp radius constants are given. [ABSTRACT FROM AUTHOR]

Abstract: Given angles , we discuss various extremal problems over the class of polynomials endowed with the norm [Copyright &y& Elsevier]

We show that Spector's 'restricted' form of bar recursion is sufficient (over system T) to define Spector's search functional. This new result is then used to show that Spector's restricted form of bar recursion is in fact as general as the supposedly more general form of bar recursion. Given that these two forms of bar recursion correspond to the (explicitly controlled) iterated products of selection function and quantifiers, it follows that this iterated product of selection functions is T-equivalent to the corresponding iterated product of quantifiers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present analytical characterizations of Mazurʼs intersection property (MIP), the CIP and the MIP⁎ via a specific class of convex functions and their conjugates. More precisely, let X be a Banach space and be its dual. Then X has the MIP if and only if for every extended real-valued lower semi-continuous convex function f defined on X with bounded domain, f is the supremum of all functions of the form: for some and , . And X has the CIP if and only if for every extended real-valued lower semi-continuous convex function on X with relatively compact domain, is the infimum of all functions which are of the form: [Copyright &y& Elsevier]

Abstract: Let be the class of functions which are analytic with and in the open unit disk . For such functions , the subclass of is introduced. The object of the present paper is to discuss some properties for in the class concerned with some sufficient conditions for to be univalent in . [Copyright &y& Elsevier]

Abstract: We define the Cartan–Hartogs domain, which is the Hartogs type domain constructed over the product of bounded Hermitian symmetric domains and compute the explicit form of the Bergman kernel for the Cartan–Hartogs domain using the virtual Bergman kernel. As the main contribution of this paper, we show that the main part of the explicit form of the Bergman kernel is a polynomial whose coefficients are combinations of Stirling numbers of the second kind. Using this observation, as an application, we give an algorithmic procedure to determine the condition that their Bergman kernel functions have zeros. [Copyright &y& Elsevier]

Abstract: The family of subsets of is 2-union free if it does not contain three different sets A, B, C such that , and is union free if it does not contain different sets , , …, such that . Let , be the maximum size of a union free, union free family, respectively. Simple construction shows that , which slightly improves the lower bound obtained by Kleitman if n is odd. Another more complicated construction further improves this lower bound. Union free family of m subsets of naturally corresponds to an matrix with independent rows. The result is related to maximum number m of independent rows in some matrix with n columns. [Copyright &y& Elsevier]

We introduce an integrodifferential operator Js,b(f) which plays an important role in the Geometric Function Theory. Some theorems in differential subordination for Js,b(f) are used. Applications in Analytic Number Theory are also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function. [ABSTRACT FROM AUTHOR]